# Proth's theorem

In number theory, Proth's theorem is a primality test for Proth numbers.

It states[1][2] that if p is a Proth number, of the form k2n + 1 with k odd and k < 2n, and if there exists an integer a for which

${\displaystyle a^{\frac {p-1}{2}}\equiv -1\mod {p},}$

then p is prime. In this case p is called a Proth prime. This is a practical test because if p is prime, any chosen a has about a 50 percent chance of working.

If a is a quadratic nonresidue modulo p then the converse is also true, and the test is conclusive. Such an a may be found by iterating a over small primes and computing the Jacobi symbol until:

${\displaystyle \left({\frac {a}{p}}\right)=-1.}$

Thus, in contrast to many Monte Carlo primality tests (randomized algorithms that can return a false positive), the primality testing algorithm based on Proth's theorem is a Las Vegas algorithm, always returning the correct answer but with a running time that varies randomly.

## Numerical examples

Examples of the theorem include:

• for p = 3 = 1(21) + 1, we have that 2(3-1)/2 + 1 = 3 is divisible by 3, so 3 is prime.
• for p = 5 = 1(22) + 1, we have that 3(5-1)/2 + 1 = 10 is divisible by 5, so 5 is prime.
• for p = 13 = 3(22) + 1, we have that 5(13-1)/2 + 1 = 15626 is divisible by 13, so 13 is prime.
• for p = 9, which is not prime, there is no a such that a(9-1)/2 + 1 is divisible by 9.

The first Proth primes are (sequence A080076 in the OEIS):

3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153 ….

The largest known Proth prime as of 2016 is ${\displaystyle 10223\cdot 2^{31172165}+1}$ , and is 9,383,761 digits long.[3] It was found by Szabolcs Peter in the PrimeGrid distributed computing project which announced it on 6 November 2016.[4] It is also the largest known non-Mersenne prime and largest Colbert number.[5] The second largest known Proth prime is ${\displaystyle 19249\cdot 2^{13018586}+1}$ , found by Seventeen or Bust.[6]

## Proof

The proof for this theorem uses the Pocklington-Lehmer primality test, and closely resembles the proof of Pépin's test. The proof can be found on page 52 of the book by Ribenboim in the references.

## History

François Proth (1852–1879) published the theorem in 1878[7][8]